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pro vyhledávání: '"Berndt, Bruce"'
Autor:
Berndt, Bruce C., Moree, Pieter
Ramanujan, in his famous first letter to Hardy, claimed a very precise estimate for the number of integers that can be written as a sum of two squares. Far less well-known is that he also made further claims of a similar nature for the non-divisibili
Externí odkaz:
http://arxiv.org/abs/2409.03428
Several methods are used to evaluate finite trigonometric sums. In each case, either the sum had not previously been evaluated, or it had been evaluated, but only by analytic means, e.g., by complex analysis or modular transformation formulas. We est
Externí odkaz:
http://arxiv.org/abs/2403.03445
Autor:
Berndt, Bruce C., Rebák, Örs
On page 206 in his lost notebook, Ramanujan recorded an incomplete septic theta function identity. Motivated by the completion of this identity by the second author, we offer cubic and quintic analogues. Using the theory generated by these two analog
Externí odkaz:
http://arxiv.org/abs/2312.15513
The modified Bessel functions $K_{\nu}(z)$, or, for brevity, K-Bessel functions, arise at key places in analytic number theory. In particular, they appear in beautiful arithmetic identities. A survey of these arithmetical identities and their appeara
Externí odkaz:
http://arxiv.org/abs/2303.02520
Publikováno v:
Advances in Mathematics, 423 (2023),109041, ISSN 0001-8708
We provide a proof of a conjecture made by Richard McIntosh in 1996 on the values of the Franel integrals, $$\int_0^1((ax))((bx))((cx))((ex))\,dx,$$ where $((x))$ is the first periodic Bernoulli function. Secondly, we extend our ideas to prove a simi
Externí odkaz:
http://arxiv.org/abs/2211.06504
Two classes of finite trigonometric sums, each involving only $\sin$'s, are evaluated in closed form. The previous and original proofs arise from Ramanujan's theta functions and modular equations.
Comment: 10 pages
Comment: 10 pages
Externí odkaz:
http://arxiv.org/abs/2210.04659
We evaluate in closed form several classes of finite trigonometric sums. Two general methods are used. The first is new and involves sums of roots of unity. The second uses contour integration and extends a previous method used by two of the authors.
Externí odkaz:
http://arxiv.org/abs/2210.00180
We consider two sequences $a(n)$ and $b(n)$, $1\leq n<\infty$, generated by Dirichlet series $$\sum_{n=1}^{\infty}\frac{a(n)}{\lambda_n^{s}}\qquad\text{and}\qquad \sum_{n=1}^{\infty}\frac{b(n)}{\mu_n^{s}},$$ satisfying a familiar functional equation
Externí odkaz:
http://arxiv.org/abs/2204.09887